Gradient flows of the entropy for jump processes

Matthias Erbar

Displaying similar documents to “Gradient flows of the entropy for jump processes”

A local approach to $g$ -entropy

Mehdi Rahimi (2015)

Kybernetika

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In this paper, a local approach to the concept of $g$ -entropy is presented. Applying the Choquet‘s representation Theorem, the introduced concept is stated in terms of $g$ -entropy.

Operator entropy inequalities

M. S. Moslehian, F. Mirzapour, A. Morassaei (2013)

Colloquium Mathematicae

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We investigate a notion of relative operator entropy, which develops the theory started by J. I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341-348]. For two finite sequences A = (A₁,...,Aₙ) and B = (B₁,...,Bₙ) of positive operators acting on a Hilbert space, a real number q and an operator monotone function f we extend the concept of entropy by setting $S_{q}^{f} (A | B) : = \sum_{j = 1}^{n} A_{j}^{1 / 2} {(A_{j}^{- 1 / 2} B_{j} A_{j}^{- 1 / 2})}^{q} f (A_{j}^{- 1 / 2} B_{j} A_{j}^{- 1 / 2}) A_{j}^{1 / 2}$ , and then give upper and lower bounds for $S_{q}^{f} (A | B)$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004),...

Symbolic extensions for nonuniformly entropy expanding maps

David Burguet (2010)

Colloquium Mathematicae

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A nonuniformly entropy expanding map is any ¹ map defined on a compact manifold whose ergodic measures with positive entropy have only nonnegative Lyapunov exponents. We prove that a $^{r}$ nonuniformly entropy expanding map T with r > 1 has a symbolic extension and we give an explicit upper bound of the symbolic extension entropy in terms of the positive Lyapunov exponents by following the approach of T. Downarowicz and A. Maass [Invent. Math. 176 (2009)].

Orders of accumulation of entropy

David Burguet, Kevin McGoff (2012)

Fundamenta Mathematicae

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For a continuous map T of a compact metrizable space X with finite topological entropy, the order of accumulation of entropy of T is a countable ordinal that arises in the context of entropy structures and symbolic extensions. We show that every countable ordinal is realized as the order of accumulation of some dynamical system. Our proof relies on functional analysis of metrizable Choquet simplices and a realization theorem of Downarowicz and Serafin. Further, if M is a metrizable Choquet...

The entropy conjecture for diffeomorphisms away from tangencies

Gang Liao, Marcelo Viana, Jiagang Yang (2013)

Journal of the European Mathematical Society

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We prove that every $C^{1}$ diffeomorphism away from homoclinic tangencies is entropy expansive, with locally uniform expansivity constant. Consequently, such diffeomorphisms satisfy Shub’s entropy conjecture: the entropy is bounded from below by the spectral radius in homology. Moreover, they admit principal symbolic extensions, and the topological entropy and metrical entropy vary continuously with the map. In contrast, generic diffeomorphisms with persistent tangencies are not entropy expansive. ...

Sequence entropy and rigid σ-algebras

Alvaro Coronel, Alejandro Maass, Song Shao (2009)

Studia Mathematica

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We study relationships between sequence entropy and the Kronecker and rigid algebras. Let (Y,,ν,T) be a factor of a measure-theoretical dynamical system (X,,μ,T) and S be a sequence of positive integers with positive upper density. We prove there exists a subsequence A ⊆ S such that $h_{μ}^{A} (T, ξ |) = H_{μ} (ξ | (X | Y))$ for all finite partitions ξ, where (X|Y) is the Kronecker algebra over . A similar result holds for rigid algebras over . As an application, we characterize compact, rigid and mixing extensions via relative...

Entropy solutions for nonhomogeneous anisotropic $Δ_{p ⃗ (\cdot)}$ problems

Elhoussine Azroul, Abdelkrim Barbara, Mohamed Badr Benboubker, Hassane Hjiaj (2014)

Applicationes Mathematicae

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We study a class of anisotropic nonlinear elliptic equations with variable exponent p⃗(·) growth. We obtain the existence of entropy solutions by using the truncation technique and some a priori estimates.

On Pawlak's problem concerning entropy of almost continuous functions

Tomasz Natkaniec, Piotr Szuca (2010)

Colloquium Mathematicae

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We prove that if f: → is Darboux and has a point of prime period different from $2^{i}$ , i = 0,1,..., then the entropy of f is positive. On the other hand, for every set A ⊂ ℕ with 1 ∈ A there is an almost continuous (in the sense of Stallings) function f: → with positive entropy for which the set Per(f) of prime periods of all periodic points is equal to A.

Continuous dependence of the entropy solution of general parabolic equation

Mohamed Maliki (2006)

Annales de la faculté des sciences de Toulouse Mathématiques

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We consider the general parabolic equation : $u_{t} - Δ b (u) + d i v F (u) = f$ in $Q =] 0, T [\times ℝ^{N}, T > 0$ with $u_{0} \in L^{\infty} (ℝ^{N}),$ $for a . e t \in] 0, T [, f (t) \in L^{\infty} (ℝ^{N})$ and $\int_{0}^{T} {∥f (t)∥}_{L^{\infty} (ℝ^{N})} d t < \infty .$ We prove the continuous dependence of the entropy solution with respect to $F,$ $b,$ $f$ and the initial data $u_{0}$ of the associated Cauchy problem. This type of solution was introduced and studied in [MT3]. We start by recalling the definition of weak solution and entropy solution. By applying an abstract result (Theorem...

On the branching property of entropy

Inder Jeet Taneja (1977)

Annales Polonici Mathematici

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The topological entropy versus level sets for interval maps (part II)

Jozef Bobok (2005)

Studia Mathematica

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Let f: [a,b] → [a,b] be a continuous function of the compact real interval such that (i) $c a r d f^{- 1} (y) \geq 2$ for every y ∈ [a,b]; (ii) for some m ∈ ∞,2,3,... there is a countable set L ⊂ [a,b] such that $c a r d f^{- 1} (y) \geq m$ for every y ∈ [a,b]∖L. We show that the topological entropy of f is greater than or equal to log m. This generalizes our previous result for m = 2.

Entropy of a doubly stochastic Markov operator and of its shift on the space of trajectories

Paulina Frej (2012)

Colloquium Mathematicae

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We define the space of trajectories of a doubly stochastic operator on L¹(X,μ) as a shift space $(X^{ℕ}, ν, σ)$ , where ν is a probability measure defined as in the Ionescu-Tulcea theorem and σ is the shift transformation. We study connections between the entropy of a doubly stochastic operator and the entropy of the shift on the space of trajectories of this operator.

On quantum informational thermodynamics with macrostates defined with respect to several operators

Bolesław Szafnicki

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CONTENTSINTRODUCTION.................................................................................................................................................................................. 5Chapter 1. PRELIMINARIES.............................................................................................................................................................. 7§ 1.1. Basic concepts of quantum mechanics.................................................................................................................................

A disjointness theorem involving topological entropy

François Blanchard (1993)

Bulletin de la Société Mathématique de France

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Margulis Lemma, entropy and free products

Filippo Cerocchi (2014)

Annales de l’institut Fourier

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We prove a Margulis’ Lemma Besson-Courtois-Gallot, for manifolds whose fundamental group is a nontrivial free product $A * B$ , without 2-torsion. Moreover, if $A * B$ is torsion-free we give a lower bound for the homotopy systole in terms of upper bounds on the diameter and the volume-entropy. We also provide examples and counterexamples showing the optimality of our assumption. Finally we give two applications of this result: a finiteness theorem and a volume estimate for reducible manifolds. ...

A strongly degenerate quasilinear equation : the elliptic case

Fuensanta Andreu, Vicent Caselles, José Mazón (2004)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We prove existence and uniqueness of entropy solutions for the Neumann problem for the quasilinear elliptic equation $u - div 𝐚 (u, D u) = v$ , where $v \in L^{1}$ , $𝐚 (z, ξ) = \nabla_{ξ} f (z, ξ)$ , and $f$ is a convex function of $ξ$ with linear growth as $∥ ξ ∥ \to \infty$ , satisfying other additional assumptions. In particular, this class includes the case where $f (z, ξ) = ϕ (z) ψ (ξ)$ , $ϕ > 0$ , $ψ$ being a convex function with linear growth as $∥ ξ ∥ \to \infty$ . In the second part of this work, using Crandall-Ligget’s iteration scheme, this result will permit us to prove existence and uniqueness of entropy solutions...